Lagged Variables

Obtain the mean lag and the long and short run multipliers for the following distributed lag models. 

Is there any problem with ordinary least squares Using the method you have described, fit the model above to the data in Table F5.1. Report your results. 

Because the model has both lagged dependent variables and autocorrelated disturbances, ordinary least squares will be inconsistent. Consistent estimates could be obtained by the method of instrumental variables. We can use x and Xt 2 as the instruments for yy and yt.2. Efficient estimates can be obtained by a two step procedure. We write the model asy, - pyM = a(l-p) + P(xy - pxM) + y(y, i - py, 2) + 5(yt 2 - pty3) + With a consistent estimator of p, we could use FGLS. The residuals from the IV estimator can be used to estimate p. Then OLS using the transformed data is asymptotically equivalent to GLS. The method of Hatanaka discussed in the text is another possibility. 

Using the real consumption and real disposable income data in Table F5.1, we obtain the following results Estimated standard errors are shown in parentheses. (The estimated autocorrelation based on the IV estimates is. 9172.) All three sets of estimates are based on the last 201 observations. 1950.4 to 2000.4 

Show how to estimate a polynomial distributed lag model with lags of 6 periods and a third order polynomial using restricted least squares. 

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To include information about process dynamics, lagged variables can be included in X. The (auto)correlograms of all x variables should be developed to determine first how many lagged values are relevant for each variable. Then the data matrix should be augmented accordingly and used to determine the principal components that will be used in the regression step. 

We use AR models to denote the time-lagged variables within the X matrix, which is represented as... 

Note 7—Rigid adherence to the prescribed heating rate is essential to reproducibility of results. Either a gas burner or electric heater may be used, but the latter must be of the low-lag, variable output type to maintain the prescribed rate of heating. 

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

This eompensation is intended for voltage-mode eontrolled forward eonverters whieh exliibit a seeond order output filter pole eharaeteristie. This would also inelude a quasi-resonant forward-mode eonverter that uses variable frequency, voltage-mode control. The T-C filter has a severe 180 degree phase lag and a -40dB/decade gain rolloff. To get any sort of wide bandwidth from the supply at all, this type of compensation must be used. 

Cohen and Coon observed that the response of most uncontrolled (controller disconnected) processes to a step change in the manipulated variable is a sigmoidally shaped curve. This can be modelled approximately by a first-order system with time lag Tl, as given by the intersection of the tangent through the inflection point with the time axis (Fig. 2.34). The theoretical values of the controller settings obtained by the analysis of this system are summarised in Table 2.2. The model parameters for a step change A to be used with this table are calculated as follows... 

We may note that the coefficient D is not zero, meaning that with a lead-lag element, an input can have instantaneous effect on the output. Thus while the state variable x has zero initial condition, it is not necessarily so with the output y. This analysis explains the mystery with the inverse transform of this transfer function in Eq. (3-49) on page 3-15. 

This configuration is also referred to as interacting PID, series PID, or rate-before-reset. To eliminate derivative kick, the derivative lead-lag element is implemented on the measured (controlled) variable in the feedback loop. 

The advantage of using the time lag method is that the partition coefficient K can be determined simultaneously. However, the accuracy of this approach may be limited if the membrane swells. With D determined by Eq. (12) and the steady-state permeation rate measured experimentally, K can be calculated by Eq. (10). In the case of a variable D(c ), equations have been derived for the time lag [6,7], However, this requires that the functional dependence of D on Ci be known. Details of this approach have been discussed by Meares [7], The characteristics of systems in which permeation occurs only by diffusion can be summarized as follows ... 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

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